Suppose that $f(\theta)$ is a $2\pi$ periodic function with a known Fourier series. Let $\alpha$ be a real number, and let $g(\theta) = f(\theta - \alpha), \theta \in \mathbb{R}$. Find the Fourier series for g.
Im having trouble figuring out this one, any help would be appreciated
Just work through the definitions.
$$\hat g(k) = \int_0^{2\pi} g(x)e^{-2\pi i x k} \ dx = \int_0^{2\pi} f(x-a) e^{-2\pi i x k}.$$
Here we make the change the variables $y=x-a$. We can keep the limits of integration the same because of the periodicity of $f$.
$$\int_0^{2\pi} f(x-a) e^{-2\pi i x k}\ dx=\int_0^{2\pi} f(y) e^{-2\pi i y k}e^{-2\pi i a k} \ dy=e^{-2\pi i a k}\int_0^{2\pi} f(y) e^{-2\pi i y k} \ dy=e^{-2\pi i a k}\hat f(k).$$